Two invariant surface-tensors determine CSL of massive body wave function
TTwo invariant surface-tensors determine CSL of massive body wave function
Lajos Di´osi ∗ Wigner Research Center for PhysicsH-1525 Budapest 114, P.O.Box 49, Hungary (Dated: August 7, 2019)Decoherence of massive body wave function under Continuous Spontaneous Localization isreconsidered. It is shown for homogeneous probes with wave functions narrow in position andangle that decoherence is a surface effect. Corresponding new surface integrals are derived asthe main result. Probe’s constant density and two completely geometric surface-dependentinvariant tensors encode full dependence of positional and angular decoherence of masses,irrespective of their microscopic structure. The two surface-tensors offer a new insight intoCSL and a flexible approach to design laboratory test masses.
I. INTRODUCTION
Spontaneous decoherence and collapse models, reviewed e.g. by [1, 2] share the form of modifiedvon Neumann equation of motion for the quantum state ˆ ρ : d ˆ ρdt = − i (cid:126) [ ˆ H, ˆ ρ ] + D ˆ ρ, (1)where ˆ H is the many-body Hamiltonian of masses m a with positions ˆx a and momenta ˆp a , resp.,for a = 1 , , . . . . The term of spontaneous decoherence takes this generic form: D ˆ ρ = − (cid:90) (cid:90) D ( r − r (cid:48) )[ ˆ (cid:37) ( r ) , [ ˆ (cid:37) ( r (cid:48) ) , ˆ ρ ]]d r d r (cid:48) , (2)containing the mass density operator at location r :ˆ (cid:37) ( r ) = (cid:88) a m a δ ( r − ˆx a ) . (3)The non-negative decoherence kernel D ( r − r (cid:48) ) is model dependent. In a conference talk [4], Icompared some characteristic features of the two leading proposals, the CSL of Ghirardi, Pearle,and Rimini, and the DP-model of Penrose and myself [5, 6]. I visualized some observations on CSLin Fig. 1 that have been waiting for mathematical formulation until now. In recent literature, the ∗ Electronic address: [email protected] a r X i v : . [ qu a n t - ph ] A ug horizontal position4x strongerposition, angleposition, angle FIG. 1: For a generic shape, both position & angle decohere (left). For a sphere, angle does not decohere(middle). N perpendicular gaps enhance longitudinal decoherence by a factor about (N+1) (right). central mathematical object is the geometric factor of decoherence: µ k = (cid:88) a m a e − i kr a , (4)introduced by [7], also discussed by [8] in this volume. This object is the Fourier-transform of theclassical mass density in the c.o.m. frame: µ ( r ) = (cid:88) a m a δ ( r − r a ) . (5)Usually, the contribution of the geometric factor is evaluated in the Fourier-representation. I amgoing to show that working in the physical space instead of Fourier’s is not only possible but evendesirable.In Sec. II we recapitulate the decoherence of c.o.m. motion in terms of the geometric factor.For constant density probes, Sec. III derives a new practical expression of the decoherence interms of a simple surface integral, the method is applied for angular (rotational) decoherence inSec. IV. Possible generalizations towards probes with unsharp edges and for wider superpositionsare outlined in Sec. IV, while Sec. V is for conclusion and outlook. II. CENTER-OF-MASS DECOHERENCE
The CSL model introduces two universal parameters, collapse rate λ = 10 − s − , localization σ = 10 − cm , and it contains the nuclear mass m N . The decoherence kernel D ( r − r (cid:48) ) is a Gaussianwhose nonlocal effect can be absorbed by a Gaussian smoothening of the mass density ˆ (cid:37) ( r ). Thekey quantity is the σ -smoothened mass distribution operator:ˆ (cid:37) σ ( r ) = (cid:88) a m a G σ ( r − ˆx a ) , (6)where G σ ( r ) is the central symmetric Gaussian distribution of width σ . Then the decoherenceterm (2) becomes a single-integral: D ˆ ρ = − π / λσ m N (cid:90) [ ˆ (cid:37) σ ( r ) , [ ˆ (cid:37) σ ( r ) , ˆ ρ ]]d r . (7)Inserting Eq. (6), Fourier-representation yields this equivalent form: D ˆ ρ = − λσ π / m N (cid:90) e − k σ (cid:88) a,b m a m b [e i kˆx a , [e − i kˆx b , ˆ ρ ]]d k . (8)We are interested in the c.o.m. dynamics of the total mass M = (cid:80) a m a : d ˆ ρ cm dt = − i (cid:126) [ ˆ H cm , ˆ ρ cm ] + D cm ˆ ρ cm , (9)where ˆX , ˆP will stand for the c.o.m. coordinate and momentum. To derive the c.o.m. decoherenceterm, substitute ˆx a = ˆX + r a in (8), where r a are the equilibrium values of the constituentcoordinates in the c.o.m. frame. Then Eq. (8) reduces to the following c.o.m. decoherence term: D cm ˆ ρ cm = − λσ π / m N (cid:90) e − k σ | µ k | (cid:16) e i k ˆX ˆ ρ cm e − i k ˆX − ˆ ρ cm (cid:17) d k , (10)where we recognize the presence of the geometric factor µ k . At small quantum uncertainties, when | ∆ X | (cid:28) σ , we use the momentum-diffusion equation as a good approximation: D cm ˆ ρ cm = − λσ π / m N (cid:90) e − k σ | µ k | [ k ˆX , [ k ˆX , ˆ ρ cm ]]d k . (11) III. INVARIANT SURFACE-TENSOR FOR C.O.M. DECOHERENCE
As we see, the geometric factor µ k itself does not matter but its squared modulus does. Weconsider the approximation (11) which allows for a spectacular simple geometric interpretation ofthe relevant structure (cid:90) e − k σ | µ k | ( k ◦ k )d k = (2 π ) (cid:90) ∇ µ σ ( r ) ◦ ∇ µ σ ( r )d r . (12)We can recognize µ σ ( r ) as the σ -smoothened mass density in the c.o.m. frame. This latter formbecomes amazingly useful if the bulk is much larger than σ and possesses constant density (cid:37) whenaveraged over the scale of σ . If, furthermore, we assume the density drops sharply from (cid:37) to zerothrough the surface then ∇ µ σ ( r ) is vanishing everywhere but in about a σ -layer around the surface.Let n stand for the normal vector of the surface at a given point r and let h be the height abovethe surface, then ∇ µ σ ( r + h n ) = − (cid:37) n g σ ( h ) , (13) g σ ( h ) is the central Gaussian of width σ . The volume integral can be rewritten, with good approx-imation, as an integral along h and a subsequent surface integral:(2 π ) (cid:90) ∇ µ σ ( r ) ◦ ∇ µ σ ( r )d r = (2 π ) (cid:37) (cid:73) n ◦ n (cid:18)(cid:90) g σ ( h )d h (cid:19) d S = (2 π ) (cid:37) π / σ (cid:73) ( n ◦ n )d S. (14)Using Eqs. (12) and (14), the decoherence term (11) obtains the attractive form D cm ˆ ρ cm = − πλσ (cid:37) m N (cid:73) [ n ˆX , [ n ˆX , ˆ ρ cm ]]d S. (15) This is our main result.
It shows that the c.o.m. decoherence is completly determined by theconstant density (cid:37) and the shape of the body, through the surface-tensor
S =: (cid:73) ( n ◦ n )d S. (16)In CSL, the c.o.m. decoherence of homogeneous bulks is a surface effect !Observe that the main result (15) remains valid if the probe has cavities in it. This allows us tomultiply the CSL decoherence By carving cavities inside the otherwise homogeneous probe, CSLdecoherence can be multipled (cf. Fig. 1). This explains the reason of enhanced decoherence inlayered structures, proposed by [9].The heating rate Γ cm = D cm ( ˆP / M ) of the c.o.m. motion is now easy to write in a moreexplicite form than before. Reading D cm off from (15), one immediately obtainsΓ cm = 2 πλσ (cid:37) m N SM = 2 πλσ (cid:37)m N SV , (17)where S is the total surface (including cavities’ internal surfaces) and V is the total volume (exclud-ing cavities). Note that Γ cm is the same if we start from the general dynamics (10) not restrictedby | ∆ X | (cid:28) σ . Interestingly, c.o.m. heating is inverse proportional to the size of the bulk. Recallthe total heating rate Γ = D (cid:88) a ˆp m a = 3 (cid:126) λ m N σ M, (18)always much larger than the c.o.m. heating. For a sphere of radius R we get Γ cm / Γ = 3( σ/R ) . Examples.
Consider the longitudinal motion of a cylinder, Eq. (15) reduces to D cm ˆ ρ cm = − πλσ (cid:37) m N S ⊥ [ˆ x, [ˆ x, ˆ ρ cm ]] , (19)where S ⊥ is the total surface perpendicular to the motion (i.e.: the area of both faces of the cylin-der). At a given constant density (cid:37) , the decoherence is independent of the length of the cylinder. Itcan be squeezed to become a plate or elongated to become a rod. This invariance of the decoherenceoffers a fair guidance when we design laboratory probes. However, the same invariance may raiseconceptual questions as well. With increasing length of the rod while decoherence rate remainsconstant, CSL might leave the longitudinal superposition of our massive rod with counter-intuitivelong coherence times. An other remarkable feature of the surface-tensor S is that spontaneousdecoherence in one direction can be decreased by tilted edges instead of perpendicular ones. If thefaces of the cylinder are replaced by cones of apex angle θ then spontaneous longitudinal deco-herence becomes suppressed by the factor sin( θ/ IV. ROTATIONAL DECOHERENCE
Our main result (15) on decoherence of lateral superpositions tells us how to calculate decoher-ence of angular superpositions. It turns out that rotational decoherence, too, is a surface effect.Let us consider small rotations around a single axis n rot for convenience. The small lateral dis-placement n ˆX − (cid:104) n ˆX (cid:105) —effective in (15)— will be replaced by the small rotational displacement n ( r × n rot )( ˆ ϕ − (cid:104) ˆ ϕ (cid:105) ) where ˆ ϕ is the angle of rotation. Then, with the scalar triple product notation,the main equation (15) reads: D rotcm ˆ ρ cm = − πλσ (cid:37) m N (cid:73) [ r , n , n rot ] d S [ ˆ ϕ, [ ˆ ϕ, ˆ ρ cm ]] . (20)Rotational decoherence is determined by the constant density (cid:37) and the rotational surface-tensor :S rot =: (cid:73) ( r × n ) ◦ ( r × n )d S. (21)Remember, our starting equation (15) was valid for | ∆ X | (cid:28) σ only, hence the validity of (20)requests the corresponding smallness of the angular uncertainties.Calculation of the spontaneous heating rate of the rotational degrees of freedom is straightfor-ward, yielding Γ rot = 2 πλσ (cid:37)m N Tr(I − S rot ) , (22)where I = (cid:82) ( r ◦ r )d r is the inertia tensor of the probe. Examples.
Consider the rotation of a long cylindric rod of length L and radius R (cid:28) L , arounda perpendicular axis through its center. All along the rod —except for its short middle part of size ∼ R — the expression [ r , n , n rot ] = r sin(Φ) is a good approximation where r ∈ ( − L/ , L/
2) is theaxial coordinate and Φ is the azimuthal angle of the surface position r . Using this approximation,we can easily evaluate the axial element of the rotational surface-tensor S rot that controls theangular decoherence (20): (cid:73) [ r , n , n rot ] d f = πRL . (23)As another example, consider our cylinder rotating around its axis of symmetry: CSL predictszero decoherence (cf. Fig. 1). But we introduce a small elliptical excentricity e (cid:28) r , n , n rot ] = Re sin(2Φ), yielding the following contributionof the shape to the strength of angular decoherence: (cid:73) [ r , n , n rot ] d f = e πR L, (24)that is e / e = 2∆ R/R where ∆ R is the smalldifference between the main diameters of the elliptic cross section. The obtained result mayraise the same conceptual problem that we mentioned for the longitudinal superposition of themassive rod/needle: azimuthal superpositions of massive cylinders of low excentricity may becomepractically insensitive to CSL. V. OUTLINES OF GENERALIZATIONS
That in CSL the c.o.m and rotational decoherences are surface effects for homogeneous probeshas been explicitly shown in Secs. III and IV for ideal sharp edges and for spatial superpositionsmuch smaller than σ . Both of the latter restrictions can be relaxed and D cm still remains a surfaceintegral.The case of unsharp edges is not much different from the ideal case. If the profile H ( h ) (cid:37) of howthe density drops from the constant (cid:37) down to zero through a thin layer defining the surface wherethe layer’s thickness is small w.r.t. the sizes of the probe then the following generalization of Eq.(13) helps: ∇ µ σ ( r + h n ) = (cid:37) n (cid:90) g σ ( h − h (cid:48) )d H ( h (cid:48) ) . (25)The rest of constructing the surface integral is the same as for Eq. (13) which described the specialcase where H was the (descending) step function.The case of not necessarily small quantum positional and angular quantum uncertainties wasdescribed by Eq. (10). It takes an equivalent closed form in coordinate representation: D cm ˆ ρ cm ( X , Y ) = − λσ π / m N (2 π ) (cid:90) (cid:2) µ σ ( r + X ) µ σ ( r + Y ) − µ σ ( r ) (cid:3) d r ˆ ρ cm ( X , Y ) . (26)The relevant structure is the integral, which we write as(2 π ) (cid:90) [ µ σ ( r + X − Y ) − µ σ ( r )] µ σ ( r )d r . (27)As long as the quantum uncertainty | X − Y | is much smaller than the sizes of the probe, theintegral is vanishing everywhere in the bulk except for a thin layer of thickness ∼ | X − Y | aroundthe surface. Accordingly, CSL decoherence remains a surface effect and, investing some hardermathematical work, D cm as well as D rot would take a form of surface integral, generalizing (15)and (20) beyond their quadratic approximations in ˆX and ˆ ϕ . VI. CONCLUDING REMARKS
We have discussed CSL for constant density test masses and proved that spontaneous deco-herence of both translational and rotational motion is determined by the density (cid:37) and by twoinvariant surface-tensors of the bodies: S = (cid:73) ( n ◦ n )d S, S rot = (cid:73) ( r × n ) ◦ ( r × n )d S. These two fully encode the relevant features of the probe’s geometry. Previously, these featureswere encoded by the so-called geometric factor µ k = (cid:37) (cid:90) e − i kr d r , an integral over the probe’s volume and a function of the wave number k . In case of general heavilyinhomogeneous test masses the necessity of using the geometric factor is certainly doubtless. Butfor homogeneous probes, the surface-tensors should take over the role.Important is the new insight into the physics of CSL in motion of a general massive bulk asa whole. First, microscopic structure is totally irrelevant, only the σ -smoothened density mat-ters. Furthermore, displacements of homogeneous regions are not decohered at all. Only thedisplacements of inhomogeneities are decohered. The sharper the inhomogenity, the stronger thedecoherence it induces. In a constant density probe, the only inhomogeneous part is its surface,hence is CSL decoherence a surface effect for it — that we have here exploited. But surface inhomo-geneity is a sharpest possible one, and decoherence for probes with smooth inside inhomogeneitiesis likely to remain dominated by the surface, our method of surface-tensors might remain valid forthem! Layer inhomogeneities with thin walls between them are competitive, their effect is surfaceeffect and our surface-tensors could be generalized to include them. Whether and when lower thantwo-dimensional inhomogeneities could play a role — that worth investigation.The author thanks the National Research Development and Innovation Office of HungaryProjects Nos. 2017-1.2.1-NKP-2017-00001 and K12435, and the EU COST Action CA15220 forsupport. [1] A. Bassi and G.C. Ghirardi, Phys. Rep. , 257 (2003)[2] A. Bassi , K. Lochan, S. Satin, T.P. Singh, H. Ulbricht H,
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